ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  ineqan12d GIF version

Theorem ineqan12d 3204
Description: Equality deduction for intersection of two classes. (Contributed by NM, 7-Feb-2007.)
Hypotheses
Ref Expression
ineq1d.1 (𝜑𝐴 = 𝐵)
ineqan12d.2 (𝜓𝐶 = 𝐷)
Assertion
Ref Expression
ineqan12d ((𝜑𝜓) → (𝐴𝐶) = (𝐵𝐷))

Proof of Theorem ineqan12d
StepHypRef Expression
1 ineq1d.1 . 2 (𝜑𝐴 = 𝐵)
2 ineqan12d.2 . 2 (𝜓𝐶 = 𝐷)
3 ineq12 3197 . 2 ((𝐴 = 𝐵𝐶 = 𝐷) → (𝐴𝐶) = (𝐵𝐷))
41, 2, 3syl2an 284 1 ((𝜑𝜓) → (𝐴𝐶) = (𝐵𝐷))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103   = wceq 1290  cin 2999
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071
This theorem depends on definitions:  df-bi 116  df-tru 1293  df-nf 1396  df-sb 1694  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-v 2622  df-in 3006
This theorem is referenced by:  fvun1  5383  fndmin  5420  offval  5877  ofrfval  5878  offval3  5919
  Copyright terms: Public domain W3C validator